Let $m$ be an integer bigger than one, $A$ a ring of algebraic integers,
$F$ its fraction field, and $K_m (A)$ the $m$-th Quillen $K$-group of $A$.
We give a (huge) explicit bound for the order of the torsion subgroup
of $K_m (A)$ (up to small primes), in terms of $m$, the degree of $F$ over
$\mathbf Q$, and its absolute discriminant.